Unusual One-Armed Density Waves in the Cassini Division of Saturn's Rings

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Division of Saturn’s Rings

Richard French, Colleen Mcghee-French, Philip Nicholson, Mathew Hedman, Nicole Rappaport, Essam Marouf, Pierre-Yves Longaretti, Joe Hahn

To cite this version:

Richard French, Colleen Mcghee-French, Philip Nicholson, Mathew Hedman, Nicole Rappaport, et

al.. Unusual One-Armed Density Waves in the Cassini Division of Saturn’s Rings. Icarus, Elsevier,

2020, 339, pp.113600. �10.1016/j.icarus.2019.113600�. �hal-02404628�

TEX

Unusual One-Armed Density Waves in the Cassini Division of Saturn’s Rings

Richard G. French, Colleen A. McGhee-French, ¹ Philip D. Nicholson, ² Mathew M. Hedman, ³ Nicole J. Rappaport, ⁴ Essam A. Marouf, ⁵

Pierre-Yves Longaretti, ⁶ and Joe Hahn ⁷

1

Department of Astronomy, Wellesley College, Wellesley MA 02481

2

Department of Astronomy, Cornell University, Ithaca NY 14853

3

Department of Physics, University of Idaho, Moscow, ID 83844

4

Cassini Radio Science Team, Jet Propulsion Laboratory (retired), Pasadena, CA 91109

5

San Jose State University, San Jose, CA 95192

6

Institute of Planetology and Astrophysics of Grenoble, France

7

Space Science Institute, Cedar Park, TX 78613

ABSTRACT

We identify several unusual one-armed density waves in Saturn’s Cassini Division, using occultation observations from the Cassini RSS and VIMS instruments. In the inner Cassini Division, we identify outward-propagating density waves with wavenum- ber m = 1 located near 118,050 km (W118.05), 118,400 km (W118.40), and 118,530 km (W118.53). From Cassini ISS images taken at very low incidence angles we find evidence for vertical structure in these waves, which may be evidence for splashing of ring material in the crests of the waves. We propose that these waves are driven by nearby eccentric ringlets, probably the Strange (or R6) and Herschel ringlets. In the outer Cassini Division, we identify a wave structure near 120,200 km (W120.20), be- tween the Laplace and Bessel gaps, as a conspicuous example of a standing wave in a planetary ring system. It appears to be the superposition of an outward-propagating m = 1 density wave, probably driven by the nearby Laplace ringlet, and its reflection at the inner edge of the Bessel gap.

Keywords: occultations, planets: rings

1. INTRODUCTION

Corresponding author: Richard G. French

rfrench@wellesley.edu

Over a decade of Cassini stellar and radio occultation experiments have pro- vided ample data for investigations of organized, km-scale structure in Saturn’s rings in ways that were previously impossible. This work has included studies of satellite-driven density and bending waves in the C ring, B ring and Cassini Division (Colwell et al. 2009a; Bailli´e et al. 2011; Nicholson and Hedman 2016;

Hedman and Nicholson 2016); non-circular ringlets and gap edges (Nicholson et al.

2014a,b; French et al. 2016a), many of which are perturbed by satellite resonances or free normal modes; and waves in the C ring driven by internal oscillations in Saturn or by quasi-permanent gravity anomalies within the planet (Hedman and Nicholson 2013, 2014; French et al. 2016b, 2019; Hedman et al. 2019). In this same period, studies of large sequences of Cassini imaging data have made it possible to character- ize the non-circular outer edges of the A and B rings (Spitale and Porco 2009, 2010;

El Moutamid et al. 2016), as well as the perturbed edges of the Encke and Keeler gaps in the outer A ring (Tiscareno et al. 2005; Weiss et al. 2009; Tajeddine et al.

2017) and a large number of satellite-driven density and bending waves, chiefly in the A ring (Tiscareno and Harris 2018).

The Cassini Division, located between the bright, optically thick A and B rings, is one of the most dynamically complex regions of Saturn’s rings. With an average normal optical depth τ ≃ 0.1 and a relatively low single particle albedo ̟ 0 ≃ 0.25 (Cooke 1991), it more closely resembles the C ring than either the A or B rings (Cuzzi et al. 1984; Esposito et al. 1984). Overviews of the Cassini Division and its salient features are shown in a Cassini image in Fig. 1 and in a radial occultation profile in Fig. 2. Although only 4500 km wide, this region is host to no fewer than eight narrow gaps (Colwell et al. 2009b). Its inner edge is defined by the outer edge of the B ring, long known to be perturbed by the 2:1 Inner Lindblad Resonance (ILR) with the satellite Mimas (Goldreich and Tremaine 1978; Porco et al. 1984) but now recognized also to be the site of several additional normal modes with azimuthal wavenumbers ranging from m = 1 to m = 5 and radial amplitudes between 5.6 and 37.0 km (Spitale and Porco 2010; Nicholson et al. 2014a).

Initial studies of other gap edges in this region using stellar occultation data from

the Visual and Infrared Mapping Spectrometer (VIMS) instrument on Cassini as

well as radio occultations carried out with the spacecraft’s Radio Science Subsystem

(RSS) revealed that while most have circular outer edges, almost all of the inner gap

edges are measurably eccentric (Hedman et al. 2010; French et al. 2010). A detailed

study of the edges of all eight gaps as well as the four dense ringlets within the

Cassini Division by French et al. (2016a) utilized the full suite of stellar occultation

data from VIMS and the Cassini Ultraviolet Imaging Spectrometer (UVIS), as well

as an expanded set of RSS data. In addition to confirming the above results, this

study revealed the existence of a large number of normal edge-modes, with m-values

ranging from 0 to 13 and radial amplitudes between 0.20 and 7.6 km.

- B ring

- Huygens gap - Huygens ringlet

- Strange ringlet (R6) - W118.05

- Herschel ringlet - W118.40

- W118.53 - Russell gap

- Jeffreys gap

- Kuiper gap

- Laplace gap - Laplace ringlet - W120.20 - Bessel gap - Barnard gap

- Triple band

Figure 1. An overview of the Cassini Division with its named gaps and ringlets, from Cassini image N1495286873 (taken on 2005 May 20). The four waves that are the subject of this investigation are identified as W118.05, W118.40, W118.53 and W120.20. The projected ring plane resolution of the image is 4 km/pixel and the observer ring elevation is −21.6 ^◦ .

Besides the quasi-regular pattern of gaps with their inner/outer edge eccentric- ity patterns, the Cassini Division harbors at least seven narrow ringlets, four of which have optical depths equal to or exceeding that of the surrounding material.

(The remaining three are faint, tenuous features generally detectable only in images (Porco et al. 2005; Colwell et al. 2009b); they do not appear in Fig. 2.) The narrow, opaque Huygens ringlet, located within the eponymous gap, was found by Turtle et al.

(1991), Spitale et al. (2006) and Spitale and Hahn (2016) to exhibit both m = 1 ( i.e.,

elliptical) and m = 2 perturbations, with the latter being attributed to the nearby

117.6 117.8 118.0 118.2 118.4 118.6 118.8 119.0 Radius (1000 km)

-0.1 0.0 0.1 0.2 0.3 0.4

Normal Optical Depth

B ring edge Huygens gap Band 1 Herschel gap Band 2 Russell gap Band 3 Jeffreys gap W118.05

W118.40 W118.53

←Huygens ringlet

Strange ringlet (R6) Herschel ringlet Pan 6:5 ILR

Prom. 9:7 ILR

119.0 119.2 119.4 119.6 119.8 120.0 120.2 120.4

Radius (1000 km) -0.1

0.0 0.1 0.2 0.3 0.4

Normal Optical Depth

Kuiper gap Laplace gap

Bessel gapBarnard gap W120.20

Laplace ringlet→

Figure 2. Prominent ringlets and gaps in the Cassini Division, as seen in an optical depth profile from the Cassini RSS egress occultation on rev 7. The wave features that are the focus of this work are illustrated: W118.05, W118.40, W118.53, and W120.20. The data are shown at a radial processing resolution of 1 km.

Mimas 2:1 resonance. Also located within the Huygens gap is the very narrow R6 ringlet (sometimes referred to informally as the Strange ringlet, a moniker we adopt here as well). Although less than 2 km in width, this ringlet shows a wide variety of normal modes with 1 ≤ m ≤ 5 as well as a significant inclination (a sin i = 7.4 km) with respect to the mean ring plane (French et al. 2016a).

Unlike the Huygens and R6 ringlets, the Herschel ringlet is comparable in optical depth to the surrounding material. In addition to showing several weak normal modes, its edges are both eccentric and inclined, although the pericenters of the inner and outer edges are not closely aligned (French et al. 2016a). Outermost of the narrow ringlets is the relatively opaque Laplace ringlet, which is eccentric but apparently not inclined.

In contrast to the gap and ringlet edges, there have been relatively few studies of the

regions of the Cassini Division between the gaps, several of which exhibit significant

internal structure. In this paper, we explore the properties of three regions of wavelike

structure in the Cassini Division, as labelled in Fig. 2. First, we examine the structure

in two broad bands in the inner Cassini Division, henceforth referred to as Bands

1 and 2. This region was first investigated in detail by Marouf and Tyler (1986), who interpreted the observed fluctuations in optical depth seen in the single Voyager radio occultation profile as moonlet wakes, similar to those seen near the Encke gap (Showalter et al. 1986). They proposed that the wakes would be produced by a pair of unseen nearby satellites in the Herschel gap. In order to explain the relatively long wavelengths and large amplitudes of these structures, however, it was necessary to postulate that the satellites in question were located within 10 − 15 ^◦ of longitude of the occultation trace, and were of order 10 km in size.

In a subsequent analysis of Voyager imaging (ISS), radio occultation (RSS), and stellar occultation (PPS) data, Flynn and Cuzzi (1989) found instead that the wave- like structures in Bands 1 and 2, specifically those that we label below and in Fig. 2 as W118.05 and W118.40, were essentially axisymmetric, meaning that their radial wavelengths showed no significant variation with longitude, such as would be expected for moonlet wakes. Furthermore, they could find no evidence in the images for waves at the edges of the Herschel gap, such as those seen on the edges of the Encke and Keeler gaps in the A ring (Cuzzi and Scargle 1985), or for variations in radial wave- length across individual images. They therefore concluded that the moonlet wake model was very unlikely to be correct. Indeed, no such satellites were ever discovered in the Voyager images, nor have any been seen in the much more extensive Cassini imaging data set, despite several searches sensitive to km-size bodies (Nicholson et al.

2018).

Flynn and Cuzzi (1989) did, however, suggest an alternative interpretation of the structure in Band 1. Noting that both edges of this region showed evidence for radial perturbations with m = 1 (i.e., they could be fit by freely precessing ellipses), and that the outward decrease in wavelength was consistent with that of a density wave with wavenumber m = 1, they proposed that the wave in Band 1 might well be an outward-propagating density wave. Such a model implied a value for the local background surface mass density of ∼ 1 g cm ⁻² , consistent with more recent determinations for the Cassini Division (Colwell et al. 2009a). Although they were unable to suggest a source for the wave, Flynn and Cuzzi (1989) also noted that the implied resonance radius was close to the apoapse of the eccentric Huygens ringlet.

Using a much larger set of Cassini observations, we confirm that both Bands 1 and 2 are dominated by outward-propagating density waves with m = 1. We further speculate that the origins of these waves are to be found in perturbations from the Huygens (or possibly Strange) and Herschel ringlets.

Second, we identify a somewhat less-prominent wavelike structure in the outer

Cassini Division between the Laplace and Bessel gaps as a conspicuous example of a

long-wavelength standing wave in a planetary ring system. It appears to be the result

of the coaddition of an outward-propagating m = 1 density wave, perhaps driven

by the nearby Laplace ringlet, and its reflection at the inner edge of the Bessel gap.

To our knowledge, this feature has not previously been discussed in print, although Borderies et al. (1985) predicted the presence of a standing wave inside the dense B ring, using a granular flow model and Spitale and Porco (2010) interpreted complex m = 1 structure seen in Cassini images interior to the outer B ring edge as evidence of a standing wave.

We organize our results as follows. In Section 2, we present a brief overview of the observations used in our investigation and describe the appearance and regional context of the waves that are the focus of our study. We review our techniques for identifying wave properties in Section 3, and Section 4 contains our investigation of the wave structure in the inner Cassini Division, in what we refer to above as Bands 1 and 2. From measurements of individual wave crests and from wavelet analysis of the overall wave structure, we identify two prominent m = 1 ILR-type waves, and explore possible wave generation mechanisms. In Section 5 we investigate, but ultimately reject, the possibility that these waves are bending rather than density waves. In Section 6, we characterize the standing wave in the outer Cassini Division, and describe the circumstances required for standing waves to exist in the rings.

Finally, in Section 7, we summarize our results and pose a series of unanswered questions to be addressed in the future.

2. OVERVIEW OF THE REGIONS STUDIED

Figure 2 shows an occultation profile of the Cassini Division, with the individual gaps and ringlets labeled. ¹ The regions studied in the present paper are those between the Huygens and Herschel gaps, referred to as Band 1 by Flynn and Cuzzi (1989);

that between the Herschel and Russell gaps, referred to by them as Band 2; and the unnamed ramp-shaped region between the Laplace and Bessel gaps. For the sake of brevity, and to be consistent with the notation applied to waves in the C ring by Colwell et al. (2009b), Hedman and Nicholson (2013) and later authors, we will refer to the waves in these three regions by their mean radii in thousands of km, as W118.05, W118.40/W118.53 and W120.20, respectively. (The waves in the inner and outer parts of Band 2 have somewhat different characteristics, so we assign them individual names.)

Of the three regions studied here, the structure in Band 1, bounded by the Huygens and Herschel gaps and located between radii of 117,930 and 118,190 km, is the most complex. Figure 3 shows a set of RSS occultation profiles of this region, offset

1

The eight gaps have been officially named by the IAU Committee on Solar System Nomenclature; we

adopt the same names for their embedded ringlets, following the practice of French et al. (2016a).

vertically for clarity and sorted and labeled by true anomaly ² , where we have assumed a local mean apsidal precession rate of ˙ ̟ = 5.000 ^◦ d ⁻¹ , with a corresponding orbital radius of ∼117,957 km. Note that this rate also does a reasonable job of displaying the eccentric inner edge of the Herschel gap, whose measured precession rate is ˙ ̟ = 4.9736 ^◦ d ⁻¹ (French et al. 2016a). Starting at the inner edge of Band 1, and extending out to ∼ 118, 080 km, we see a series of six or seven smooth, low-amplitude oscillations with an average wavelength of ∼ 15 km. The systematic progression of peak locations to smaller radii with increasing true anomaly (cf. Fig. 8) is indicative of a trailing spiral pattern. This is the first region studied by Marouf and Tyler (1986) and by Flynn and Cuzzi (1989).

The outer third of Band 1, however, exhibits a more irregular profile with a series of sharp peaks whose structure varies systematically as a function of the local true anomaly, but whose geometry is neither clearly trailing nor leading. Some isolated spikes in optical depth, such as those at ∼ 118, 090 km and ∼ 118, 150 km, are prominent only over a limited range in true anomaly. Separating these two portions of Band 1 is the relatively weak density wave driven by the Prometheus 9:7 ILR at ∼ 118, 066 km (Colwell et al. 2009b) (labeled in Fig. 2), though this wave is not well-resolved at the 1 km resolution of the profiles in Fig. 3. In addition to the 10–

20-km scale wavelike structure seen over most of Band 1, which is the subject of our study, there are zones of much finer-scale structure near both inner and outer edges, with wavelengths as short as 1-2 km. The oscillations near the outer edge, between 118,170 and 118,190 km, were interpreted by Flynn and Cuzzi (1989) as a possible wake driven by a nearby km-size moonlet in the Herschel gap.

Figure 4 shows the same set of RSS occultation profiles for Band 2, again sorted by true anomaly, but here with ˙ ̟ = 4.950 ^◦ d ⁻¹ , with a corresponding orbital radius of ∼118,289 km. Band 2 is bounded by the Herschel and Russell gaps and located between radii of 118,283 and 118,590 km. The eccentric inner edge of the Russell gap has a measured precession rate of ˙ ̟ = 4.9092 ^◦ d ⁻¹ (French et al. 2016a) and is readily seen at the right edge of the figure. The structure in Band 2 is somewhat simpler than that in Band 1. The inner half of the region — which we identify here as wave W118.40 — shows a series of seven or eight regular undulations with an average wavelength of ∼ 10 km, while the outer half — here designated as W118.53

— shows a set of ∼ 15 higher-frequency oscillations. In the inner zone the pattern of peaks in optical depth clearly form a trailing spiral pattern, similar to that in Band 1 – slanting dotted vertical lines to guide the eye approximately follow the decreasing orbital radius with increasing true anomaly, for several adjacent wavecrests. Also as in Band 1, these two sub-regions are separated by a weak density wave; in this case

2

The true anomaly f is the angular distance of the occultation ring intercept point from the nominal

pericenter of a precessing, eccentric orbit at the local distance from Saturn. In terms of the inertial

longitude λ and observation time t, it is given by f = λ− λ

− ̟(t ˙ −t

), where ˙ ̟ is the local apsidal

precession rate, t

is the reference epoch of 2008 Jan 1.5 UTC., and λ

is the inertial longitude of

periapse at the epoch.

W118.05 Region

-100 -50 0 50 100 150 200

r - 118000 km 0

5 10 15

Normal Optical Depth

f= 3.9 3.9 9.9 38.3 59.1 70.2 90.0 91.4 91.7 105.5 165.4 173.9 174.1 178.9 180.0 189.9 193.4 197.9 214.1 236.2 262.8 269.0 273.9 276.2 279.2 296.1 300.8 317.8 322.3 338.5 358.0

Figure 3. Optical depth profiles of wave W118.05 in Band 1, offset vertically for clarity

and sorted by true anomaly f (labeled at right) for an assumed pattern speed of 5.000 ^◦ d ⁻¹ .

The data are from X-band RSS occultations on revs 7 – 67, processed at a radial resolution

of 1 km. The narrow R6 (or Strange) ringlet and the outer edge of the Huygens gap are

seen at the left edge of the plot, while the eccentric inner edge of the Herschel gap is seen at

the right edge. The latter has a substantial radial amplitude of ae = 8.3 km. Only barely

visible at this scale and resolution is the weak Prometheus 9:7 ILR density wave outward

of its resonance radius of ∼ 118, 075 km, marked by a vertical dashed line.

due to the Pan 6:5 ILR at ∼ 118, 453 km (Colwell et al. 2009b). (The wave itself is unresolved at the resolution of Fig. 4, but appears as a prominent peak in optical depth at ∼ 118, 465 km, as labeled in Fig. 2.) Across Band 2, the average radial wavelength decreases smoothly from ∼ 12 km at the inner edge to ∼ 5 km at the outer edge. The studies by Marouf and Tyler (1986) and Flynn and Cuzzi (1989) both focussed on the structure in the inner half of Band 2, between radii of 118,350 and 118,450 km.

The third of our wave features, W120.20, is located in the band between the Laplace and Bessel gaps, between radii of 120,086 and 120,231 km, in a region where the background optical depth is strongly decreasing. Figure 5 shows a sequence of RSS occultation profiles of this region, again sorted by true anomaly, this time with

˙

̟ = 4.720 ^◦ d ⁻¹ , with a corresponding orbital radius of ∼119,872 km. Visible at the right edge of the figure is the eccentric inner edge of the Bessel gap, which has a measured precession rate of ˙ ̟ = 4.6845 ^◦ d ⁻¹ , as determined from fits to multiple occultation measurements (French et al. 2016a). The eccentricity is difficult to see at the scale of this figure. Wavelike structure is only apparent to the eye in the outer half of the band, outside ∼ 120, 180 km, but wavelet analysis (see Section 6) clearly shows that it extends inward at least to 120,140 km. Just as for the inner region of the W118.05 wave (117,970 - 118,060 km) and the W118.40/W118.53 complex, the radial wavelength of the W120.20 wave clearly decreases outwards, in this case from perhaps 25 km at the inner edge of the band to ∼ 5 km at the outer edge. But unlike the waves in Bands 1 and 2, the W120.20 wave is neither clearly trailing nor leading, a curious fact to which we will return in Section 6.1. This wave does not seem to have been the subject of any previous published studies.

3. IDENTIFYING WAVELIKE STRUCTURE

We begin by assuming that each wave is a spiral density wave with an azimuthal wavenumber m, rotating relative to inertial space at an angular velocity or pattern speed Ω p . As long as such a wave is weak enough to remain in the linear regime, the perturbed surface density can be written as the real part of

σ(r, λ, t) = σ ₀

1 + A(r)e ^i(φ

^(r)+φ

⁾

, (1)

where σ 0 is the background surface mass density and A(r) is a slowly-varying am- plitude factor. The phase of the wave is composed of both a rapidly-varying radial function φ r (r) that describes the oscillatory part of the wave, plus a more slowly- varying azimuthal phase factor given by (Shu 1984)

φ λ,t = |m|[λ − λ 0 − Ω p (t − t 0 )]. (2)

Here, λ and t are the longitude and time at which the occultation track crossed the

wave, t 0 is a reference epoch (= 2008 Jan 1.5 UTC), and λ 0 is a reference longitude.

W118.40 and W118.53

-100 -50 0 50 100 150 200

r - 118400 km 0

5 10 15

Normal Optical Depth

f= 14.5 39.0 45.8 45.9 57.8 62.1 76.7 90.7 133.2 146.7 149.3 168.5 173.4 175.8 185.4 191.9 219.1 221.2 232.1 233.2 237.5 258.4 262.1 267.3 296.8 307.6 309.3 318.9 327.3 339.8

Figure 4. Optical depth profiles of waves W118.40 and W118.53 in Band 2, offset vertically

for clarity and sorted by true anomaly f (labeled at right) for an assumed pattern speed

of 4.950 ^◦ d ⁻¹ . The data are from X-band RSS occultations on revs 7 – 67, processed at

a radial resolution of 1 km. The eccentric inner edge of the Russell gap, with a radial

amplitude ae = 7.6 km, is seen at the extreme right edge of the plot. The slanted dotted

lines approximately trace the trend of decreasing radius with increasing true anomaly of

several wavecrests in the inner region of W118.40. Visible as a narrow peak in optical depth

separating the two wave regions is the Pan 6:5 ILR density wave at ∼ 118, 465 km, marked

by a vertical dashed line.

W120.2 Region

100 120 140 160 180 200 220 240

r - 120000 km 0

5 10 15

Normal Optical Depth

f= 1.6 21.9 29.3 35.5 42.3 63.4 76.1 85.5 87.6 106.4 111.8 170.4 182.4 184.6 185.2 194.5 198.1 203.0 207.8 215.1 242.3 243.5 260.7 277.7 285.4 288.3 300.1 308.0 345.9 350.5

Figure 5. Optical depth profiles of wave W120.20 in the outer Cassini Division, offset vertically for clarity and sorted by true anomaly f (labeled at right) for an assumed pattern speed of 4.720 ^◦ d ⁻¹ . The data are from X-band RSS occultations on revs 7 – 67, again processed at a radial resolution of 1 km. The Bessel gap and its eccentric inner edge, with a radial amplitude of ae = 1.8 km, is seen at the right edge of the plot.

With this formulation the radial phase φ r of such a wave is assumed to be the same

at all longitudes and times. As in our previous work (Hedman and Nicholson 2013,

2014; French et al. 2016a), we denote ILR-type density waves by positive values of

m and Outer Lindblad Resonance (OLR-type) waves by negative values of m. With

this convention, the pattern speeds of both types of wave are given by the common expression

mΩ _p = (m − 1)n + ˙ ̟ _sec , (3)

where n is the local Keplerian mean motion and ˙ ̟ sec is the apsidal precession rate due to secular perturbations (primarily Saturn’s J 2 and J 4 ). For waves with m = 1, such as those encountered here, we have Ω p = ˙ ̟ sec . For a wave with a specified pattern speed and m-value, the resonant radius is the location where Eq. (3) is satisfied exactly.

At some distance from the resonance, the radial wavenumber k(r) = dφ r /dr for freely-propagating density waves is given by the asymptotic expression (Shu 1984;

Hedman and Nicholson 2013):

|k(r)| ≃ [3(m − 1) + ²¹ ₂ J 2 (R p /r L ) ² ]M p

2πσ 0 r _L ⁴ (r − r L ), (4)

where r L is the resonant radius, M p is the mass of the central planet, J 2 is the zonal gravity harmonic, and R _p is the reference radius for J ₂ . ILR-type density waves (m > 0) propagate outwards from an ILR, while OLR-type density waves (m ≤ 0) propagate inwards from an OLR. In both cases, |k| increases linearly with distance from r L . Referring back to Eqns. (1) and (2), we see that k(r) > 0 corresponds to a trailing spiral wave (i.e., the radius of a wave crest decreases with increasing longitude) while k(r) < 0 for a leading spiral, a fact we will make use of in Section 6.1 below. The corresponding radial phase is given by

φ _r (r) − φ ₀ = Z r

r

k(r ^′ )dr ^′ ≃ ± 1

2 K (r − r _L ) ² , (5) where φ ₀ is a constant, K refers to the constant factor in Eq. (4), and the ± sign describes trailing and leading waves, respectively. For m = 1, the factor of 3(m − 1) in the expression for k(r) is zero and in this case the radial wavelength is significantly longer than for other values of m. ³

In general, our goal is to combine data from multiple occultations to determine both the azimuthal wavenumber m and the pattern speed Ω p for each wave, and then to compare these with the above theoretical expressions. In order to identify the nature of the wavelike structure such as that in Bands 1 and 2, we first convert each occultation-derived optical depth profile into a wavelet profile, using the same proce- dure described by Hedman and Nicholson (2013, 2014) to analyze unknown waves in the C ring. We denote the wavelet transform of profile i by W _i (r, k), where r is radius and k is the radial wavenumber. W i is a complex quantity, with both an amplitude

3

This is because, as noted above, the pattern speed is much lower for waves with m = 1 and the

ring’s self-gravity is only required to compensate for the radial gradient in the apsidal precession

rate ˙ ̟ rather than for that in the mean motion n.

A _i (r, k) and a phase Φ _i (r, k). We use two methods in the current work. In the first, which follows that employed by Hedman and Nicholson (2013, 2014), French et al.

(2016b) and French et al. (2019), we compare the difference in the measured phase for pairs of observations δΦ i,j = Φ i − Φ j with the difference predicted by Eq. (2), which we denote as δφ i,j

δφ i,j = |m|[δλ − Ω p δt], (6) where δλ and δt are the differences in longitude and time between the two occultation profiles at the radius of interest. This procedure involves subtracting the phases of the two wavelet transforms, W _i and W _j , so that their implicit common radial dependence described by φ _r cancels out, and then averaging over specified ranges of r and k. For further details the reader is referred to the above papers. We then sum the squares of δΦ i,j − δφ i,j over all distinct pairs of occultations and scan over a range of values of Ω p to find the pattern speed corresponding to the minimum RMS phase difference for a specified value of m. Plotting the RMS value of δΦ i,j − δφ i,j vs Ω p provides a useful visual way to judge the significance of the best-fitting model. In general, only a single value of m will yield an acceptably small value of the RMS phase difference.

In the second method, as developed by Hedman and Nicholson (2016) and also employed by Hedman et al. (2019) and Hedman and Nicholson (2019), the wavelet transform for each occultation is first corrected to zero longitude and to the adopted epoch using trial values of m and Ω _p via the expression

W φ,j (r, k) = W j e ^−iφ

= A j (r, k) e ^i(Φ

^−φ

⁾ , (7) where we denote the predicted wave phase φ λ,t by φ j . These phase-corrected wavelets are then summed over all occultations to obtain an average wavelet, as if all observa- tions had been made at zero longitude and at time t 0

hW _φ (r, k )i = 1 N

X

j

W _φ,j (r, k), (8)

where N is the number of occultations combined. In this case, the assumed pattern speed is varied until a maximum in the power of the average phase-corrected wavelet is obtained for the specified value of m

P φ (r, k) = |hW φ i| ² =

1 N

X

j

W φ,j (r, k)

2

. (9)

Finally, this phase-corrected power is divided by the average uncorrected wavelet power

P ¯ (r, k) = h|W j | ² i = 1 N

X

j

|W j (r, k)| ² . (10)

It can be shown (Hedman and Nicholson 2016) that this normalized power, R(r, k) =

P φ (r, k)/ P ¯ (r, k) lies between 0 and 1, depending on the accuracy of Ω p and whether or

not multiple waves are present. Plotting the normalized power vs both r and k shows both the radial extent and the varying radial wavenumber of the wave, while plotting the power P _φ (r, k) averaged over k vs r and Ω _p provides an idea of the uncertainty in pattern speed and the probable location of the Lindblad resonance driving the wave, i.e., the location at which Eq. (3) is satisfied.

One advantage of this procedure is that, for the wrong value of Ω p and given many occultations, the average wavelet and its power are both close to zero. A second advantage is that, unlike the first procedure, it can handle situations where more than one wave is present at the same radius. A final advantage is that the combined phase-corrected wavelet may be inverted to obtain an average profile of the wave, with the phase differences between individual occultation profiles removed.

In the following sections, we will employ both of these procedures in order to de- termine the best-fitting values of m and Ω p for each wave, using different subsets of occultation data.

4. WAVE STRUCTURE IN THE INNER CASSINI DIVISION

Our first step in unravelling the wavelike structure in Bands 1 and 2 was to apply the phase-corrected wavelet procedure, using a large set of VIMS stellar occultation profiles spanning the range from 2006 (R Hya, rev 36) through early 2017 (α Ori, rev 269). This is the same set of 56 occultations listed in Table 1 of French et al.

(2019). Experiments in which the azimuthal wavenumber m was varied from −10 to 10 showed significant power in the phase-corrected wavelet only for m = 1, in both regions. The results for m = 1 are shown in Fig. 6.

The successive panels in this rather complicated figure are as follows. From top to bottom, panel (i) shows a representative optical depth profile of the region, for context; here, it is the RSS egress occultation on rev 7. Panel (ii) shows the average uncorrected wavelet power ¯ P as a greyscale function of radius and radial wavenum- ber. ⁴ The most prominent features here are associated with the sharp edges of gaps and ringlets that generate significant high spatial frequency power. Panel (iii) shows the average phase-corrected wavelet power P φ , again as a greyscale function of radius and radial wavenumber. At each radius, the assumed pattern speed is that given by Eq. (3), for the specified value of m. The sharp edges are now somewhat suppressed, leaving the wave structure more obvious. Both panels are displayed on a common logarithmic scale, because of their large dynamic range. Panel (iv) shows the ratio R = P φ / P ¯ on a linear scale from 0 to 1, which highlights the fraction of the average

4

This panel is similar in format to the regional wavelet power spectra computed by

Tiscareno and Harris (2018) from high-resolution Cassini images across the entire ring system (see

their online Supplementary Material). However, the focus of their work was to identify waves asso-

ciated with known satellite resonances, and the waves we identify here are inconspicuous and not

identified in their derived radial I/F scans or wavelet spectra.

power associated with the m = 1 perturbation. Here we see significant power only in Bands 1 and 2, with wavenumbers in the range 0.2 ≤ k ≤ 0.8 km ⁻¹ , or radial wave- lengths of 7.5 − 30 km. In panel (v), we vary the assumed pattern speed (expressed here as the corresponding shift in the resonant radius, r L ) for each radius and then plot the peak value of the power ratio R within a specified range of k. For example, a maximum in the peak power at δr = −200 km at a particular radius means that the best-fitting pattern speed at that radius corresponds to a resonant radius that is 200 km less than the local value. Panel (vi) summarizes the best-fitting pattern speed and resonant radius at each location in panel (v) where the power ratio is apprecia- ble. These were obtained by fitting a 1D gaussian model to R as a function of δr, separately for each radius, and then converting the central value of r L for each wave back to a pattern speed. Dotted lines show the average values of r _L for each region.

Focussing first on the third and fourth panels in Fig. 6, we see strong evidence for m = 1 ILR-type waves in Band 1, at radii corresponding to wave W118.05, and also in Band 2 spanning waves W118.40 and W118.53. In the case of W118.05, there is a weak indication that k does increase outwards, as expected from Eq. (4). For waves W118.40 and W118.53, on the other hand, there is a clear positive gradient in k, at least between radii of 118,350 and 118,550 km. Turning now to the fifth panel, we see that the best-fitting pattern speeds are significantly faster than the local values for both regions. For W118.05, the peak power occurs for δr ≃ −150 km, or r L ≃ 117, 900 km, with an average pattern speed of 5.010 ± 0.008 ^◦ d ⁻¹ . For W118.40 and W118.53, the peak power occurs for δr ≃ −250 km, or r _L ≃ 118, 200 km, with an average pattern speed of 4.963 ± 0.008 ^◦ d ⁻¹ . The estimated uncertainties in pattern speed translate into an uncertainty in the resonant radii for these waves of

2

7 (δΩ p /Ω p )r L ≃ 50 km, since Ω p = ˙ ̟ sec ∝ r ^−7/2 for m = 1.

Although the phase-corrected wavelet analysis is the most efficient way to search a large region for significant wave signals, it is less precise than the pairwise comparison method in specifying the best-fitting pattern speed. Our next step was therefore to apply the pairwise phase comparison method separately to each of the three waves in the inner Cassini Division, under the assumption that the wave in question has m = 1. Our results are shown in Fig. 7, with a separate panel for each wave.

The W118.05 wave gives a deep minimum in the RMS phase residuals of 38.9 ^◦ at Ω _p = 5.005 ^◦ d ⁻¹ . The corresponding resonant radius is ∼117,925 km, about 37 km larger than but generally consistent with the peak-power result in Fig. 6. The radial range fitted was restricted to 117,990 – 118,090 km, where the wave signature is clearest in Fig. 3, and the range of wavelengths averaged over was set at 5 – 25 km.

Wave W118.40 yields a minimum RMS phase residual of 51.1 ^◦ at Ω p = 4.958 ^◦ d ⁻¹ .

The corresponding resonant radius is 118,232.9 km, about 30 km larger than but

generally consistent with the peak-power result in Fig. 6. The radial range fitted was

-0.1 0.0 0.1 0.2 0.3

Normal optical depth

Inner Cassini Division

W118.05

W118.40 W118.53

Band 1 Band 2 Band 3

Herschel Gap Russell Gap Jeffreys Gap

0.0 0.5 1.0 1.5

k (km

) Avg. Pwr.

0.0 0.5 1.0 1.5

k (km

)

0.0 0.5 1.0 1.5

k (km

) Pwr. of Avg. PCW

0.0 0.5 1.0 1.5

k (km

)

0.0 0.5 1.0 1.5

k (km

) Ratio

0.0 0.5 1.0 1.5

k (km

)

-1000 -500 0 500

δ r (km)

-1000 -500 0 500

δ r (km)

117.6 117.8 118.0 118.2 118.4 118.6 118.8 119.0 R (1000 km)

117.8 118.0 118.2 118.4

R

(1000 km) _R

res

= 117888 km Ω

= 5.010 deg/day

R

= 118198 km Ω

= 4.963 deg/day W118.05

W118.40 & W118.53

Figure 6. Phase-corrected wavelet analysis for Bands 1,2 and 3, assuming that the wave

structure is due to density waves with m = 1. See text for a detailed description of each

panel. Note that significant power with m = 1 is seen only in Bands 1 and 2, with the

best-fitting pattern speeds and resonant radii shown in the bottom panel.

restricted to 118,360 – 118,440 km and the range of wavelengths averaged over was set at 5 – 20 km.

The fit for wave W118.53 yields a weaker detection, with a minimum RMS phase residual of 83.8 ^◦ at Ω p = 4.964 ^◦ d ⁻¹ . The corresponding resonant radius is 118,195.2 km, very close to the peak-power result of 118,198 km in Fig. 6. The radial range fitted was 118,490 – 118,570 km and the range of wavelengths averaged over was restricted to 3 – 10 km, in an attempt to improve the fit.

For each of waves W118.05 and W118.40/118.53 the best-fitting pattern speeds from the pairwise phase comparison of δΦ i,j − δφ i,j agree with those derived from the phase-corrected wavelets to within 0.005 ^◦ d ⁻¹ , less than the estimated uncertainty in the latter values. Recalling that Ω _p = ˙ ̟ _sec ∝ r ^−7/2 for m = 1, we estimate that the differences in resonant radii are ±35 km. At this level of accuracy, the pattern speeds and resonant radii for W118.40 and W118.53 may be considered to be the same, suggesting that they are parts of a single wavetrain. We now look at the possible origins for each of these waves.

4.1. W118.05 wave

Based on the results in Figs. 6 and 7 above, this wave is provisionally identified

as an ILR-type density wave with m = 1, Ω p = 5.006 ± 0.005 ^◦ d ⁻¹ , and a resonant

radius of 117, 920 ± 35 km. This places the likely source of the wave close to the

outer edge of the Huygens gap, which has a mean radius of 117,931 km, and to the

Strange ringlet at 117,908 km (French et al. 2016a). However, neither the series of

occultation profiles in Fig. 3 nor the wavelet spectra in Fig. 6 show a very clear trend

of radial wavelength decreasing with increasing radius over the full radial extent of

the region, as expected for an ILR-type density wave. In an attempt to verify the

nature of this wave, we also measured the locations of individual wave crests in the

same sequence of RSS occultation profiles shown in Fig. 3. Although a small fraction

of our complete data set, these profiles all have a uniform and high signal-to-noise

ratio conducive to such measurements. The results are shown in Fig. 8, with each set

of measurements shown twice to facilitate linking successive crests in longitude. From

this plot we see that (i) the great majority of measured crests in the range 117,950

– 118,080 km do indeed fit a model of a trailing spiral pattern ( i.e., the radius of

each crest decreases monotonically with increasing true anomaly f = λ − ̟ ), (ii) the

spiral is an m = 1, as may be seen by observing that each ‘arm’ links to the next one

after 360 ^◦ of longitude, (iii) the radial wavelength does seem to decrease outwards,

as expected, and (iv) the scatter in the measured crest locations with respect to a

simple linear spiral is only a few km, in most cases.

W118.05

0 2 4 6 8 10

Pattern Speed (deg/day) 0

20 40 60 80 100 120

rms phase residual (deg) m = 1 ILR

Assumed r_L = 118050.0 km Predicted Speed = 4.986°/day

Minimum rms phase residual = 38.9°

Best-Fit Speed = 5.005°/day Corresponding r

L = 117924.6 km

W118.40

0 2 4 6 8 10

Pattern Speed (deg/day) 0

20 40 60 80 100 120

rms phase residual (deg) m = 1 ILR

Assumed r

L = 118400.0 km Predicted Speed = 4.933°/day

Minimum rms phase residual = 51.1°

Best-Fit Speed = 4.958°/day Corresponding r_L = 118232.9 km

W118.53

0 2 4 6 8 10

Pattern Speed (deg/day) 0

20 40 60 80 100 120

rms phase residual (deg) m = 1 ILR

Assumed r

L = 118530.0 km Predicted Speed = 4.914°/day

Minimum rms phase residual = 83.8°

Best-Fit Speed = 4.964°/day Corresponding r_L = 118195.2 km

Figure 7. Pattern speed scans for waves W118.05, W118.40 and W118.53 for m = 1, based

on comparing the observed and predicted phase differences between pairs of occultations

(see text for details). The data set includes all VIMS and RSS occultations, subject to

signal-to-noise limitations. For each wave, the best-fitting pattern speed (indicated by the

vertical dotted line) and corresponding resonant radius are listed, along with the minimum

RMS phase residual. Also listed are the predicted values of Ω _p used to center our search

region, indicated by the vertical dashed line.

However, a careful inspection of Fig. 8 also shows that the picture becomes more complicated outside a radius of ∼ 118, 080 km, corresponding roughly to the radius of the weak Prometheus 9:7 density wave (r _L = 118, 066 km). In this region we see what appear to be several partial segments of leading spirals (i.e., the radius of some crests seem to increase monotonically with increasing true anomaly). Moreover, in Fig. 3 we see in this same region a series of narrow features (e.g., the double-peak at 118,097 km and the single peaks at 118,109, 118,147 and 118,160 km) that seem to be axisymmetric rather than following the expected pattern for either type of spiral. One possible explanation, inspired by our results on the W120.20 wave described below, is that the m = 1 trailing spiral is partially reflected at the inner edge of the Herschel gap and propagates back towards the inner edge of Band 1 as a leading spiral. The result may look somewhat like an m = 1 standing wave, for which the crests move neither inward nor outward with time, but simply oscillate in place. Provocatively, Lehmann et al. (2019) (cf. Fig. 5) show a very similar result for a hydrodynamical simulation of a non-linear density wave partially reflected at a sharp discontinuity in the background optical depth.

4.2. W118.40 and W118.53

Based on Figs. 6 and 7 above, the waves in Band 2 appear to be fit best by a single ILR-type density wave with m = 1. In this case, Ω p = 4.958 ± 0.005 ^◦ d ⁻¹ and the resonant radius is 118, 230 ± 35 km. This places the likely source of the wave within or very near the Herschel ringlet, which has a mean radius of 118,249 km and an average width of 32 km (French et al. 2016a). The next-nearest feature is the inner edge of the Herschel gap at 118,188 km. Here, the series of occultation profiles in Fig. 4 and the wavelet spectra in Fig. 6 both show a clear trend of radial wavelength decreasing with increasing radius, as expected for an ILR-type density wave. Despite the interruption of this wave by a narrow, circular feature at ∼ 118, 465 km due to the density wave driven by the weak Pan 6:5 ILR (r L = 118, 453 km), the W118.40 and W118.53 waves seem to form part of a single wavetrain, with eight crests interior to 118,465 km, and ∼ 15 exterior to this radius. But based on the larger RMS phase residuals seen for W118.53, we conclude that the wave is less well-organized in this region than it is closer to the resonant radius. In this respect it may be similar to the W118.05 wave, though we have not been able to find any evidence of a reflected wave here.

4.3. A regional view of the inner Cassini Division

Figure 9 summarizes all of the measured m = 1 pattern speeds for the inner Cassini Division, drawing on the results of this paper and those of French et al. (2016a).

Sharp edges of gaps and ringlets are indicated by filled circles, while the pattern

W118.05

0 120 240 360 480 600 720

Anomaly -100

0 100 200

r-118000 km

Figure 8. Successive wave crest locations for RSS observations of wave W118.05 in Band

1, as functions of true anomaly f . Series of points for which the radius decreases with

increasing f correspond to a trailing spiral pattern, whereas points for which the radius

increases with increasing f correspond to a leading spiral. Both types of spiral are seen

here, but the trailing spiral is dominant interior to 118,090 km. Note that two full cycles

of true anomaly are shown, in order to demonstrate that, in most cases, each arm of the

trailing spiral pattern connects smoothly to the next one further inwards, as expected for

an m = 1 spiral. A total of 11 or 12 circuits of the trailing spiral are visible here, beginning

at a radius of ∼ 117, 950 km. The inferred resonant radius is 117,920±35 km.

speeds for the waves in Bands 1 and 2 are shown as open circles, with error bars. The diagonal line is the predicted apsidal precession rate ˙ ̟ _sec , based on Saturn’s zonal gravity harmonics (Jacobson et al. 2008). As noted by French et al. (2016a), almost all pattern speeds in this part of the rings are slightly higher than predicted, probably because of the unmodeled effect of the nearby and massive B ring. Particularly noticeable in this respect are the anomalously fast rates exhibited by the outer edges of the B ring itself (Nicholson et al. 2014a) and the Huygens gap, and the inner edges of the Herschel and Kuiper gaps, all of which are currently unexplained.

For each of the m = 1 waves studied here, a horizontal dotted line connects the measured pattern speed to the resonant radius, i.e., the radius where ˙ ̟ sec = Ω p . These lines reinforce the similarities in pattern speed (or apsidal precession rate) of the W118.05 wave and the Strange ringlet, and of the W118.40/W118.53 wave and both edges of the Herschel ringlet. (Although the inferred resonant radius for the W118.05 wave is also close to the outer edge of the Huygens gap, as noted above, the pattern speed matches that of the Strange ringlet much more closely. A similar statement applies to the W118.40/W118.53 wave and the inner edge of the Herschel gap.) Both waves therefore exhibit pattern speeds that are significantly faster than the local value of ˙ ̟ sec , but which match that of a nearby eccentric ringlet. Based on these facts, and on the close similarities between the two waves, we suggest that the source of the waves is likely to be the Strange (or R6) ringlet in the case of wave W118.05 in Band 1 and the Herschel ringlet in the case of wave W118.40/W118.53 in Band 2. More specifically, we propose that the ringlet eccentricities first drive forced m = 1 perturbations in the ring material adjacent to the nearby outer gap edges, which in turn generate outward-propagating m = 1 density waves. A very similar situation was studied by Hahn (2008), but in that study the eccentric ringlet was replaced by an eccentric moonlet. ⁵

There are, however, potential difficulties with this simple picture, chief of which is that the observed apsidal precession rates of the intervening gap edges — i.e., the outer edges of the Huygens and Herschel gaps — differ measurably from those of the ringlets and the density waves (see Fig. 9). Second, the inner and outer eccentric edges of the Herschel ringlet are significantly misaligned, with relatively small eccentricities (ae ∼ 1 − 2 km), raising the question of whether it could really drive an m = 1 wave, although some degree of misalignment is expected due to the viscosity of the ring (Borderies et al. 1983). A third puzzle is why the wave in Band 1 is apparently driven by the Strange ringlet rather than by the Huygens ringlet, even though the latter is both more eccentric (ae = 29 km vs 7.5 km; see Table 2 of French et al.

5

These circumstances are more similar than they might first appear, as the theory for a moonlet

driving an m = 1 density wave begins with averaging the gravitational effect of the moonlet around

its eccentric orbit, thereby effectively replacing it by an eccentric ringlet. As in the present work,

the wave predicted by Hahn (2008) is a trailing m = 1 spiral that propagated outwards from the

gap with a pattern speed equal to the apsidal precession rate of the moonlet.

Inner Cassini Division

117.5 118.0 118.5 119.0 119.5

Radius (1000 km) 4.7

4.8 4.9 5.0 5.1

Ω

(deg/day)

B ring OER

Huygens OEG Huygens OEG W118.05

W118.40

W118.53

Huygens gap Herschel gap Russell gap Jeffreys gap Kuiper gap

Huygens ringlet

Herschel ringlet

Figure 9. Regional m = 1 pattern speeds in the inner Cassini Division. Filled circles denote gap and ringlet edges from French et al. (2016a) while open circles show the waves discussed in this paper. The diagonal line shows the apsidal precession rate ˙ ̟ _sec calculated from Saturn’s zonal gravity harmonics. Horizontal dotted lines connect the measured pattern speeds for the gaps or ringlets and waves to their corresponding resonant radii. The vertical dotted line at 117570.12 km marks the mean radius of the outer edge of the B ring, from Table 4 of Nicholson et al. (2014a).

(2016a)) and probably much more massive than the former. We will return to these vexing questions in Section 7.4 below.

5. POSSIBLE VERTICAL STRUCTURE IN BANDS 1 AND 2 5.1. RSS profiles

Further investigation provides still more evidence that the picture outlined above

is not the whole story. In compiling the galleries of profiles in Figs. 3 and 4 we

observed that, although the RSS occultations all have very similar signal-to-noise

levels (unlike the stellar occultations, for which the stellar flux and ring opening angle

vary significantly between target stars), the prominence of the waves varies noticeably

between the observations. Closer inspection shows that the waves generally become

more prominent as the opening angle of the rings becomes smaller, suggesting a

vertical (i.e., out of the mean ring plane) component to the waves.

This can be tested by sorting the RSS profiles by the angle B _eff , first introduced by Gresh et al. (1986) and Nicholson et al. (1990) to characterize the effect of vertical ring structure on RSS occultation profiles. Simple geometry shows that the apparent radial displacement ∆r of a vertically-offset feature in the rings is given by ∆r =

−z/ tan B eff , where

tan B _eff = tan B ⊕

cos(λ − λ ⊕ ) . (11)

Here, λ and z are the longitude and vertical height of the feature, B ⊕ is the elevation of the Earth above the ring plane, and λ ⊕ is the saturnicentric longitude of the Earth. (A similar equation applies to stellar occultations, with B ⊙ and λ ⊕ replaced by B ∗ and λ ∗ , respectively (Jerousek et al. 2011; Nicholson and Hedman 2016).) In Figs. 10 and 11 we replot the same galleries of RSS profiles for the W118.05 and W118.40/118.53 waves, but now sorted in order of increasing |B eff |. In Fig. 11 we see that the W118.40/118.53 waves are most prominent for small values of B eff and much less obvious for |B eff | > 20 ^◦ . In the W118.05 region the waves between 118,000 and 118,060 km are also more clearly seen at |B eff | < 20 ^◦ (Fig. 10). This seems to confirm our suspicion that there may be an out-of-plane component to both waves.

As supporting evidence, we note that not only are the Strange and Herschel ringlets known to be eccentric, they also both exhibit small inclinations with respect to the mean ring plane (French et al. 2016a). It is thus conceivable that they might drive bending waves as well as density waves in the adjacent ring material, as envisioned by Hahn (2007) for an inclined satellite. As in his parallel investigation of an m = 1 density wave driven by a nearby eccentric moonlet (Hahn 2008), the theory for a moonlet driving an m = 1 bending wave begins with averaging the gravitational effect of the moonlet around its inclined orbit, thereby effectively replacing it by a circular but inclined ringlet. As discussed below, the bending wave modeled by Hahn (2007) is a leading m = 1 spiral that propagates outwards from the gap with a negative pattern speed equal to the nodal regression rate of the moonlet. Interior to the gap containing the moonlet, the bending wave is evanescent, so no similar wave is to be expected.

5.2. Bending waves?

These considerations raise the possibility that the waves in the inner Cassini Divi-

sion have some vertical component, or even that they might be bending waves rather

than density waves. But the latter possibility immediately leads to additional compli-

cations. Most bending waves — at least those generated at Inner Vertical Resonances

(IVRs) with external satellites such as the well-studied Mimas 5:3 bending wave in

the A ring (Shu et al. 1983; Gresh et al. 1986) — propagate inwards, whereas we have

seen above that the waves in question seem to propagate outwards. However, it turns

out that bending waves with m = 1 are peculiar in several respects, as first observed

W118.05 Region

-100 -50 0 50 100 150 200

r - 118000 km 0

5 10 15

Normal Optical Depth

Beff= 8.1 8.3 8.7 -10.0 10.0 -10.5 -10.8 11.2 11.3 11.5 -11.7 -12.7 -12.9 -13.0 -13.7 23.2 -41.5 -49.3 -50.8 58.2 58.4 58.7 58.7 59.7 59.7 61.8 66.4 -71.2 81.8 -86.6 88.8

Figure 10. Profiles of wave region W118.05 in Band 1, sorted by the absolute value of B _eff . The data are the same as the RSS occultation profiles shown in Fig. 3. The vertical dashed line marks the Prometheus 9:7 ILR.

by Rosen and Lissauer (1988). Not only do they have unusually long wavelengths, like m = 1 density waves, but their pattern speeds are negative. To see this, note that the analogous expression to Eq. (3) for a bending wave is:

mΩ p = (m − 1)n + ˙Ω sec , (12)

W118.40 & W118.53

-100 -50 0 50 100 150 200 250

r - 118400 km 0

5 10 15

Normal Optical Depth

Beff= 8.1 8.3 8.7 -10.0 10.0 -10.5 -10.8 11.2 11.3 11.5 -11.7 -12.7 -12.9 -13.0 -13.7 23.2 -41.5 -49.3 -50.8 58.2 58.4 58.7 58.7 59.7 61.8 66.4 -71.2 81.8 -86.6 88.8

Figure 11. Profiles of waves W118.40 and W118.53 in Band 2, sorted by the absolute value of B _eff . The data are the same as the RSS occultation profiles shown in Fig. 4. The vertical dashed line marks the Pan 6:5 ILR.

where ˙Ω sec is the nodal regression rate due to secular perturbations. For bending waves

with m = 1, we have Ω p = ˙Ω sec , which is negative for orbits about an oblate planet

such a Saturn. In fact, ˙Ω _sec ≃ − ̟ ˙ _sec . There are two known m = 1 bending waves in

Saturn’s rings, driven by the distant satellites Titan and Iapetus (Rosen and Lissauer

1988; Tiscareno et al. 2013; Nicholson and Hedman 2016). For each of these waves,

the pattern speed is indeed observed to be negative with Ω _p ≃ −n _s , where n _s is the satellite’s mean motion. As a result, these satellite-driven bending waves actually propagate outwards. Another unusual feature of m = 1 bending waves driven by IVRs is that they take the form of a leading spiral, rather than the more common trailing geometry (Rosen and Lissauer 1988; Nicholson and Hedman 2016). Similar results were obtained by Hahn (2007) in his study of m = 1 bending waves driven by a nearby inclined moonlet.

Thus, a satellite-driven (or perhaps ringlet-driven) outward-propagating m = 1 bending wave does indeed appear to be theoretically possible, albeit with a negative pattern speed and the instantaneous form of a leading spiral. If this were the case, however, we would expect to see a systematic leading or trailing pattern in the radial locations of individual wave crests, when plotted as a function of ascending node Ω.

Figure 12 reveals no such pattern for W118.05, arguing against this possibility for the wave in Band 1.

We can also test the possibility that the m = 1 waves in Bands 1 and/or 2 have a bending wave component by re-running the phase-matching wavelet analysis with the pattern speed given by Eq. (12). A practical complication is that the sign of the apparent radial displacement associated with a vertical wave changes with the sign of B eff , as given by Eq. (11), and thus with the signs of both B ⊕ and cos(λ − λ ⊕ ). This introduces additional observer-dependent 180 ^◦ shifts into φ λ,t beyond those associated with the wave itself that must be taken into account when modeling or correcting the wavelet phases. We have previously developed a simple but approximate method for adjusting the observed wavelet phases for bending waves, as described by French et al.

(2019), that involves adding 180 ^◦ to φ j whenever B eff < 0. The results are shown in Fig. 13, in the same format as Fig. 6.

Focussing on the third, fourth and fifth panels, where the phase correction is applied, we do not see strong evidence for m = 1 vertical structure in either Band 1 or Band 2.

The strongest signal is seen at a radius of 117, 700 − 118, 000 km and k ≃ 0.1 km ⁻¹ . This appears to be due to the Strange ringlet, which is known to have a substantial inclination of a sin i = 7.4 km and ˙Ω = −4.976 ^◦ d ⁻¹ (French et al. 2016a). The only sign of a possible bending wave is a rather weak signal in Band 2 with k increasing outward from ∼ 0.5 km ⁻¹ to ∼ 1.0 km ⁻¹ (12 > λ r > 6 km).

5.3. ISS images

One might ask whether there is any other evidence for vertical structure in Bands

1 and 2. Surprisingly, this seems to be the case. A search for high-resolution Cassini

images of the inner Cassini Division, using the OPUS tool on the PDS Ring-Moon

Systems Node, identified several dozen images with a projected ring plane radial res-

W118.05

0 120 240 360 480 600 720

Node Ω (deg) -100

0 100 200

r − 118000 km

Figure 12. Successive wave crest locations for RSS observations of wave W118.05 in Band 1 as in Fig. 8, but plotted as a function of ascending node Ω. There is no clear organized behavior to indicate that this is a bending wave.

olution below 2 km/pixel, some of which were obtained near the saturnian equinox

on 11 August 2009, when the sun angle on the rings was extremely low. This illu-

mination geometry is especially sensitive to vertical structure such as bending waves,

with the ring brightening whenever its local surface is tilted towards the sun. We

have analyzed five images that are particularly revealing, with parameters listed at

0.0 0.1 0.2 0.3 0.4

Normal optical depth

Inner Cassini Division - m=1 BW

0.0 0.5 1.0 1.5

k (km-1) Avg. Pwr.

0.0 0.5 1.0 1.5

k (km-1)

0.0 0.5 1.0 1.5

k (km-1) Pwr. of Avg. PCW

0.0 0.5 1.0 1.5

k (km-1)

0.0 0.5 1.0 1.5

k (km-1) Ratio

0.0 0.5 1.0 1.5

k (km-1)

117.6 117.8 118.0 118.2 118.4 118.6 118.8 119.0

-1000 -500 0 500

δr (km)

117.6 117.8 118.0 118.2 118.4 118.6 118.8 119.0

Radius (1000 km) -1000

-500 0 500

δr (km)

Figure 13. Phase-corrected wavelet analysis for Bands 1 and 2, assuming that the structure is due to bending waves with m = 1 and Ω _p ≃ ˙Ω sec . The format is the same as that of Fig. 6, except that the 6th panel is omitted here. No clear bending wave signals are seen, except the power at low wavenumbers between 117,700 and 117,950 km, which is likely due to the inclined Strange (R6) ringlet.

the top of Table 1. All are of the sunlit (i.e., southern) side of the rings. Just as for

the RSS occultations, the sensitivity of images to vertical structure is conveniently

parameterized by the effective elevation angle B eff , given by Eq. (11), but in this case

B ⊕ and λ ⊕ are replaced by the equivalent solar elevation B ⊙ and longitude λ ⊙ . In

Table 1. Cassini ISS NAC images

Image Observation name Midtime B

(deg) B

(deg) B

(deg) H(deg) km/px

Figs. 14 and 15:

N1495327885 ISS 008RI RDHRCOMP001 2005-141T00:24:09 −14.70 −21.64 52.51 287.72 1.75 N1595342033 ISS 077RI RDHRCOMP001 2008-203T13:55:42 −66.08 −5.95 6.88 329.66 1.10 N1625956787 ISS 114RI EQXSHADOW003 2009-191T21:57:58 −31.37 −0.48 −0.48 177.19 1.75 N1627303569 ISS 115RI AZSCANLIT001 2009-207T12:04:10 −50.38 −0.24 0.29 35.17 1.44 N1627307734 ISS 115RI AZSCANLIT001 2009-207T13:13:35 −48.81 −0.24 0.35 46.89 1.27

Fig. 21:

N1560311811 ISS 046RI RDHRESSCN001 2007-163T03:22:40 −33.77 −12.06 89.71 89.94 0.99 N1870073325 ISS 268RI HRRADSCNL001 2017-095T07:40:28 74.57 26.72 26.74 182.30 0.79 (a )Projected ring plane radial resolution

terms of the hour angle used by the PDS, defined as H = λ − λ ⊙ + 180 ^◦ , we have tan B _eff = − tan B ⊙

cos H . (13)

Images with small values of B eff (i.e., those with low B ⊙ and targets near the subsolar or antisolar directions, so that | cos H| is close to 1) will show alternate brightening and dimming across near-axisymmetric vertical ripples, while images with large values of B eff (i.e., those with high B ⊙ and/or targets near 90 ^◦ from the sub solar direction) are insensitive to such variations in the surface tilt.

After photometrically calibrating and then navigating each image (by correcting its nominal pointing using circular ring features with known radii), we reprojected them into radius-longitude coordinates at a common radial scale, as shown in Fig. 14. We then extracted a mean radial profile of brightness (I/F ) for each image, plotted in Fig. 15 in the same order as in Fig. 14. Vertical dashed lines in this figure show the predicted (apparent) radius for selected sharp-edged features, taking account of possible non-circularities and inclinations using the orbit models of French et al.

(2016a). From left to right, these are the Huygens and Strange ringlets, the outer edge of the Huygens gap, the Herschel gap and its embedded ringlet, and the edges of the Russell and Jeffreys gaps. In most instances, the predicted radii match those seen in the image scans, providing confirmation of our geometric reconstructions. We briefly discuss each image in turn.

N1495327885: This image was taken in 2005 on day 141, with a relatively high solar

elevation angle B ⊙ = −21.64 ^◦ . The target point was on the right ansa, ∼ 70 ^◦ from

the antisolar direction (hour angle H = 287.72 ^◦ ), a geometry that is quite insensitive

to out-of-plane structure, as is indicated by value of B eff = 52.51 ^◦ . We see that all

gaps and narrow ringlets are at their expected locations, with the only other notable

structure being a series of weak oscillations in I/F in Band 1, corresponding to wave

Inner Cassini Division

118.0 118.2 118.4 118.6 118.8 119.0

Radius (1000 km)

Jeffreys gap →

← Russell gap

← Herschel ringlet W118.05

W118.40

W118.53

Band 1 Band 2 Band 3

B

=52.51 °

B

=6.88 °

B

= − 0.48 °

B

=0.29 °

B

=0.35 °

Figure 14. Five Cassini ISS narrow angle camera images of the inner Cassini Division, reprojected at a uniform scale into radius and longitude. The effective ring opening angle B _eff is labeled at right for each image; see Table 1 for a list of image numbers and additional geometric parameters. Visible from left to right are the outer Huygens gap edge (except in the 2nd image), the Herschel gap and ringlet, the Russell gap and the Jeffreys gap. Bounded by these gaps are Bands 1, 2 and 3 of Flynn and Cuzzi (1989). The wave structures in Bands and 2 are especially prominent in the lower three images (see text).

W118.05, and even weaker oscillations in Band 2, corresponding to W118.40 and W118.53.

N1595342033: This image was taken in 2008 on day 203, with a much lower solar elevation angle B ⊙ = −5.95 ^◦ . The target point was near the edge of the shadow cast by Saturn across the rings, at hour angle H = 329.66 ^◦ , making the illumination geometry much more sensitive to out-of-plane structure, with B eff = 6.88 ^◦ . All visible gaps and narrow ringlets are again at their expected locations, but the wavelike structure in Bands 1 and 2 is now much more visible than in the first image, as we found above for the lower-inclination RSS occultation profiles. The average I/F is

∼ 0.006.

N1625956787: This image was taken in 2009 on day 191, only 1 month before the

saturnian equinox on day 223, with a solar elevation angle of −0.48 ^◦ and a target

point close to local noon (H = 177.19 ^◦ ), so that B eff = −0.48 ^◦ . Again, all of the

Inner Cassini Division

0.0 0.1 0.2 0.3 0.4

I/F

N1495327885

BSun = -21.64°

Beff = 52.51°

H = 287.72°

(a)

0.000 0.002 0.004 0.006 0.008 0.010 0.012

I/F

N1595342033

BSun = -5.95°

Beff = 6.88°

H = 329.66°

(b)

↓ ↓

0.000 0.002 0.004 0.006

I/F

N1625956787

B_Sun = -0.48°

Beff = -0.48°

H = 177.19°

(c)

→

↓ ↓ ↓

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025

I/F

N1627303569

B_Sun = -0.24°

Beff = 0.29°

H = 35.17°

(d)

→ ↓ ↓

117.8 118.0 118.2 118.4 118.6 118.8 119.0

Radius (1000 km) 0.000

0.001 0.002 0.003

I/F

N1627307734

B_Sun = -0.24°

Beff = 0.35°

H = 46.89°

(e)

→ ↓ ↓

Figure 15. Radial scans of the ISS NAC images shown in Fig. 14. See Table 1 for a list

of image numbers and geometric parameters. Visible from left to right are the Huygens

ringlet and outer gap edge, the Herschel gap and ringlet, the Russell gap and the Jeffreys

gap. Vertical dashed lines correspond to the predicted apparent radius of sharp-edged

features, taking account of their possibly non-circular and/or inclined orbital shapes. The

wave structures W118.05, W118.40 and W118.53 are especially prominent in the lower three

profiles (see text). Arrows highlight the changing features noted in the text.